Adaptivity in convolution models with partially known noise distribution

作者：Butucea Cristina^*; Matias Catherine; Pouet Christophe

来源：Electronic Journal of Statistics, 2008, 2: 897-915.

DOI：10.1214/08-EJS225

• Adaptive nonparametric tests
• convolution model
• goodness-of-fit tests
• infinitely differentiable functions
• partially known noise
• quadratic functional estimation
• Sobolev classes
• stable laws

We consider a semiparametric convolution model. We observe random variables having a distribution given by the convolution of some unknown density f and some partially known noise density g. In this work, g is assumed exponentially smooth with stable law having unknown self-similarity index s. In order to ensure identifiability of the model, we restrict our attention to polynomially smooth, Sobolev-type densities f with smoothness parameter beta. In this context, we first provide a consistent estimation procedure for s. This estimator is then plugged-into three different procedures: estimation of the unknown density f, of the functional integral f(2) and goodness-of-fit test of the hypothesis H(0) : f = f(0), where the alternative H(1) is expressed with respect to L(2)-norm (i.e. has the form psi(-2)(n) parallel to f - f(0)parallel to(2)(2) >= C). These procedures are adaptive with respect to both s and beta and attain the rates which are known optimal for known values of s and beta. As a by-product, when the noise density is known and exponentially smooth our testing procedure is optimal adaptive for testing Sobolev-type densities. The estimating procedure of is illustrated on synthetic data.

• 出版日期2008